广义修正随机梯度与广义Skorohod积分

Generalized Modified Stochastic Gradient and Generalized Skorohod Integral

应用有界算子族的加权Bochner积分,考虑连续时间Guichardet-Fock空间L^2(Γ;η)中广义修正随机梯度■h及过程空间L^2(Γ×R+;η)中的广义Skorohod积分δh,其中h是R上的非负函数,对特殊的h,相应的■h和δh恰是修正随机梯度和Skorohod积分.结果表明,■h,δh分别是L^2(Γ;η)和L^2(Γ×R+;η)中的稠定线性闭算子,一般是无界的;对于一类特殊的非负函数h,证明了相应的广义修正随机梯度■h和广义Skorohod积分δh是L^2(Γ;η)和L^2(Γ×R;η)上的有界线性算子;进一步,得到了■h,δh是关于点态修正随机梯度族{■s;s∈R+}}及其共轭族{■s^*;s∈R+}的加权Bochner积分表示,利用该表示及修正随机梯度■和Skorohod积分δ的共轭关系,得到了■h,δh的共轭关系.

We consider the generalized modified stochastic gradient■hin the continuous-time Guichardet-Fock space L^2(Γ;η)and the generalized Skorohod integralδh in L^2(Γ×R+;η)by means of the weighted-Bochner integral of the bounded linear operator,where h is a nonnegative function on R+.For a special h,■h,andδh are the modified stochastic gradient■and Skorohod integralδ.The result shows that■h are densely-defined,closed linear operators in L^2(Γ;η)and L^2(Γ×R+;η),respectively.In general,h andδh are unbounded,but for a special class of nonnegative functions h,we prove that■h andδh are bounded linear operator in L^2(Γ;η)and L^2(Γ×R+;η).Moreover,we obtain that■h andδh can be represented by the weighted-Bochner integral in terms of the point-state modified stochastic gradient{■s;s∈R+}and its adjoint operator{■s^*;s∈R+}.By using the Bochner integral of■h andδhand■h andδhare adjoint operator for each other,we obtain the adjoint relation of■h and δh.